Convergency and Divergency of Functional Coefficient Weak Instrumental Variables Models

In this paper, we consider a simultaneous equations model under a functional coefficient representation for the structural equation of interest and adopt the local-to-zero assumptions as in Staiger and Stock (1997) and Hahn and Kuersteiner (2002) on the coefficients of the instruments in the reduced form equation. Under this functional coefficient representation, models are linear in endogenous components with coefficients governed by unknown functions of the predetermined exogenous variables. We propose a two-step estimation procedure to estimate the coefficient functions. The first step is to estimate a matrix of unknown parameters of the reduced form equation based on the least squares method, and the second step is to use the local linear fitting technique to estimate coefficient functions by using the estimated reduced forms as regressors. We investigate how the limiting distribution of the proposed nonparametric estimator changes as the parameterization is allowed for different degrees of weakness. As a result, our new theoretical findings are that the possible convergency of the proposed nonparametric estimators can be attained only for the nearly weak case and the rate of convergence for the nonparametric estimator for coefficient functions of endogenous variables is slower than the conventional rate. But the nonparametric estimator for coefficient functions of endogenous variables is divergent for both the weak and nearly non-identified cases. A Monte Carlo simulation is conducted to illustrate the finite sample performance of the resulting estimator and results support these theoretical findings.